Platonic Solids

47 46 the archImedean dualS everything has its opposite The duals of the Archimedean Solids were first described as a group by Eugne Catalan 18141894 and are positioned opposite to correspond with their partners on page 33. To create the dual of an Archimedean Solid extend perpendicular lines from its edge midpoints, tangential to the Solid s midsphere. These lines are the dual s edges, the points where they first intersect each other are its vertices. Archimedean Solids have one type of vertex and different types of faces, their duals therefore have one type of face but different types of vertices. The two quasiregular Archimedean Solids, the cuboctahedron and the icosidodecahedron, both have rhombic duals which were discovered by Kepler. The Platonic dual pair compounds pages 16, 36 and 40 define the face diagonals of these rhombic polyhedra, which are in the ratios 2 for the rhombic dodecahedron and for the rhombic triacontahedron. Kepler noticed that bees terminate their hexagonal honeycomb cells with three such 2 rhombs. He also described the three dual pairs involving quasiregular solids below, where the cube is seen as a rhombic solid, and the octahedron as a quasiregular solid.